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Journal of Geophysical Research: Space Physics RESEARCH ARTICLE 10.1002/2015JA021762 Key Points: • New code of CIR energetic particle transport extends earlier analytical work • Event fitting yields the solar wind turbulence level and the seed population • Implication on particle cross-field diffusion

Correspondence to: G. Li, [email protected]

Citation: Zhao, L., G. Li, R. W. Ebert, M. A. Dayeh, M. I. Desai, G. M. Mason, Z. Wu, and Y. Chen (2016), Modeling transport of energetic particles in corotating interaction regions: A case study, J. Geophys. Res. Space Physics, 121, 77–92, doi:10.1002/2015JA021762.

Received 3 AUG 2015 Accepted 19 DEC 2015 Accepted article online 22 DEC 2015 Published online 23 JAN 2016

Modeling transport of energetic particles in corotating interaction regions: A case study Lulu Zhao1 , Gang Li1 , R. W. Ebert2 , M. A. Dayeh2 , M. I. Desai2 , G. M. Mason3 , Z. Wu4 , and Y. Chen4 1 Department of Space Science and CSPAR, University of Alabama in Huntsville, Huntsville, Alabama, USA, 2 Southwest

Research Institute, San Antonio, Texas, USA, 3 Applied Physics Laboratory, The Johns Hopkins University, Laurel, Maryland, USA, 4 Institute of Space Sciences and School of Space Science and Physics, Shandong University at Weihai, Weihai, China

Abstract We investigate energetic particle transport in corotating interaction regions (CIRs) through a case study. The CIR event we study occurred on 8 February 2008 and was observed by both the Advanced Composition Explorer (ACE) and the twin Solar TErrestrial RElations Observatory (STEREO) B spacecraft. An in situ reverse shock was observed by STEREO B (1.0 AU) but not ACE (0.98 AU). Using STEREO B observations and assuming the CIR structure does not vary significantly in the corotating frame, we estimate the shock location at later times for both the STEREO B and ACE observations. Further assuming the accelerated particle spectral shape at the shock does not vary with shock location, we calculate the particle differential intensities as observed by ACE and STEREO B at two different times by solving the focused transport equation using a Monte Carlo simulation. We assume that particles move along Parker’s field and experience no cross-field diffusion. We find that the modulation of sub-MeV/nucleon particles is significant. To obtain reasonable comparisons between the simulations and the observations by both ACE and STEREO B, one has to assume that the CIR shock can accelerate more particles at a larger heliocentric distance than at a smaller heliocentric distance.

1. Introduction Corotating interaction regions (CIRs) are formed when fast solar wind, originating from coronal holes that extend to low latitudes, compresses slow solar wind in front of it. A forward shock can form at the leading edge of the compression region and propagate into the slow solar wind, and a reverse shock can form at the trailing edge of the compression region and propagate into the fast solar wind. Early studies showed that by 3 to 4 astronomical units (AU), most CIRs are bounded by a forward and reverse shock pair [e.g., Hundhausen and Gosling, 1976]. Recently, a statistical study by Jian et al. [2006] found that 31% of CIRs observed at 1 AU are associated with shocks. CIRs are a major source of energetic particles in the inner heliosphere during solar minimum [e.g., van Hollebeke et al., 1978; McDonald et al., 1976; Richardson et al., 1993]. Fisk and Lee [1980] first examined particle acceleration associated with CIRs. They solved the steady state transport equation with a geometry appropriate to CIRs. In their model, particles are accelerated via the first-order Fermi acceleration mechanism at either the forward or the reverse shock which are often at a distance of several AU. Energetic particles then propagate along the interplanetary magnetic field back to 1 AU. Similar to galactic cosmic rays (GCRs) and anomalous cosmic rays (ACRs), the adiabatic cooling can lead to a modulation effect at low energies. Later, Giacalone et al. [2002] proposed another mechanism which, instead of invoking shocks, considered particle acceleration in gradual and slowly varying solar wind compression regions. They found that particles can be accelerated up to 10 MeV/nucleon by a process similar to diffusive shock acceleration, in which particles gain energy by scattering between converging scattering centers. Simulation results similar to observations were obtained for a reasonable set of parameters.

©2015. American Geophysical Union. All Rights Reserved.

ZHAO ET AL.

Recent observations by Mason et al. [1997] suggested that the observed energy spectra in CIR events often do not show modulation effects in low energies. Instead, the spectra continue to fall down as a power law from the energy threshold of the instrument, typically ∼30 keV/nucleon, and even solar wind energies [Chotoo et al., 2000]. More recently, Ebert et al. [2012a] studied 73 CIR-associated suprathermal He intensity enhancements and found that the peak of these sub-MeV He intensities correlate well with the arrival of the compression TRANSPORT OF ENERGETIC PARTICLES IN CIRS

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region trailing edge. These and other recent works [e.g., Mason et al., 2008a; Buˇcík et al., 2009; Ebert et al., 2012b] suggest that sub-MeV energetic particles associated with CIRs at 1 AU may be accelerated locally. While sub-MeV particles may be accelerated locally, higher-energy particles suffer less modulation and may well be accelerated beyond 1 AU (e.g., 3 to 5 AU) and propagate along Parker field back to 1 AU. Therefore, it is possible that energetic particles observed at 1 AU come from more than one radial distance. This also implies that cross-field diffusion may be important to understand CIR observations. Indeed, there is some evidence that cross-field diffusion may be important in a few CIRs [Dwyer et al., 1997]. Because cross-field diffusion can drastically reduce the distance a particle travels from several AUs to reach Earth’s orbit, it is therefore more important for low-energy particles. In this work, we study the transport of energetic particles at CIRs using a numerical model and compare our model with an earlier analytical model of Fisk and Lee [1980]. To quantify the modulation effect along a single Parker field, we assume that particles are tied to single field lines. Since there is no cross-field diffusion, particles on different field lines originate from different locations along the reverse shock. As a case study, we examine one CIR event that occurred between 8 and 14 February 2008. This event was observed by the twin Solar TErrestrial RElations Observatory (STEREO) spacecraft and the Advanced Composition Explorer (ACE) spacecraft. Observations at STEREO B (hereafter STB) showed an in situ signature of a reverse shock, which was not seen at ACE and STEREO A (hereafter STA). This in situ observation of the shock by STB allows us to estimate the shock location at a later time for ACE and STB observations upstream of the shock. We further assume that the shape of the accelerated particle spectrum at the shock does not vary with heliocentric distance. Note that this is only a working hypothesis. With this hypothesis, we can decouple the acceleration process from the transport process. This hypothesis is bound to introduce some errors in our analysis. However, as we will see, the shock in our event was quite strong at 1 AU, having a compression ratio 𝛽 =2.17. Interplanetary shocks often do not have compression ratios larger than, say, 3. The shock-accelerated particle spectrum for a shock with 𝛽 = 2.17 is ∼ p−5.5 while that with a compression ratio 𝛽 = 3.0 is ∼ p−4.5 . As we will see, the effect of transport (modulation at low energy) is much larger than the uncertainty introduced in this hypothesis. Indeed, considering the ratio of the differential intensities dJ∕dE (2 MeV)/dJ∕dE (0.2 MeV) (roughly the energy range for our problem), then the ratio of using ∼p−4.5 is larger by a factor of ∼3.2 than that using ∼p−5.5 . In comparison, the effect of modulation on this ratio is more significant. Therefore, our hypothesis, while crude, is somewhat justified. Also note that the shock location in our event is not too far from 1 AU. For the two periods we consider, it locates at 1.55 AU and 2.55 AU, respectively. The plasma parameters at these two shock locations may not differ too much from that at 1 AU and the corresponding compression ratios may be similar to 2.17. In any case, as we focus on the effects of the transport of energetic particles in this work, we here make the simplest assumption about the shape of the accelerated particle spectrum, that is that it does not vary with heliocentric distance. To examine the transport process, we use a Monte Carlo simulation. In particular, we simulate particle spectra at two different times for STB and ACE observations and compare our simulation results with observations. In our simulation, we ignore the gyration degree of freedom of the particles and solve the focused transport equation along a Parker field. Comparing to Fisk and Lee [1980], who assumed the motion of particles are diffusive in spatial coordinates and are described by a diffusion coefficient 𝜅 , we retain explicitly the particle pitch angle in our approach. Consequently, we consider explicitly the magnetic focusing effect and the pitch angle diffusion (described by D𝜇𝜇 ) in our formalism. The Fisk and Lee [1980] approach is justified when the solar wind turbulence is strong. However, when solar wind turbulence is weak, employing D𝜇𝜇 and treating the focusing effect explicitly is more appropriate. To summarize, using a case study where in situ observation of a CIR shock is available, we improve upon the Fisk and Lee model by employing a more sophisticated numerical simulation that treats the transport of CIR-associated energetic particles by a focused transport equation. Our paper is organized as follows. In section 2 we present the multispacecraft observations of this event; we discuss the simulation model in section 3 and compare the simulation results with observations in section 4. We conclude in section 5.

2. Event Analysis The energetic particle measurements we present here are from the Suprathermal Ion Telescope (SIT) [Mason et al., 2008b] on board the STA and STB spacecraft, and the Ultra-low Energy Isotope Spectrometer (ULEIS) ZHAO ET AL.

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Figure 1. CIR observations by STB, ACE, and STA (from left to right columns, respectively) between 7 February 2008 to 17 February 2008. (first row) The He intensity time profile, (second row) solar wind speed, (third row) density, (fourth row) temperature, and (fifth row) total magnetic field. The red dashed line in Figure 1 (left column) marks the shock passage at STB. The blue and green dashed lines correspond to two observation periods for which simulations were performed.

[Mason et al., 1998] on board ACE. The solar wind plasma measurements are obtained from the PLAsma and SupraThermal Ion Composition investigation (PLASTIC) [Galvin et al., 2008] on board STA and STB, and the Solar Wind Electron Proton Alpha Monitor (SWEPAM) [McComas et al., 1998] on board ACE. We also use the magnetic field data from the Magnetometers on STA and STB [Acuña et al., 2008] and ACE [Smith et al., 1998]. The CIR event we study was first observed at STB starting on 8 February 2008. It has been reported as Event 25 by Mason et al. [2009]. It has also been studied by Buˇcík et al. [2011]. Figure 1 shows the in situ observations (STB, ACE, and STA from left to right columns). Figure 1 (first row) shows the 0.14–2.2 MeV/nucleon He time intensity profiles between 7 February 2008 and 17 February 2008; Figure 1 (second row) to Figure 1 (fifth rows) show, in descending order, the solar wind proton speed, density, temperature and the total magnetic field (magnitude) in the same time period. For the STB observations, the He intensity started to increase at 18:00 UT, 8 February 2008, and reached its maximum at 19:36 UT, 9 February 2008, with a clear peak associated with it. At the time of the peak, discontinuities in the proton speed, magnetic field, and temperature were observed, indicating this time as the passage of the CIR reverse shock. This shock was identified in the STEREO shock list at http://www-ssc.igpp. ucla.edu/∼jlan/STEREO/Level3/STEREO_Level3_Shock.pdf, maintained by Dr. J. Lan. The in situ shock passage as observed by STB is shown as the red dashed line in Figure 1 (left column). The blue and green dashed lines correspond to two observation periods for which simulations were performed. The shock spectrum is obtained by integrating the differential intensity in the period 17:54 UT to 20:09 UT, 9 February and is shown ZHAO ET AL.


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by the red curve in Figure 2. In comparison, the spectrum for a later time period between 13:55 UT and 15:50 UT, 11 February, was shown as the blue curve in Figure 2. The modulation at low energies can be clearly seen for this period.

Figure 2. Differential intensity spectrum J(E) observed by STB and ACE. Red diamonds show the STB differential intensity integrated over 17:54 UT to 20:09 UT, 9 February (corresponding to the red dashed line in Figure 1). Blue diamonds show the STB differential intensity integrated over 13:55 UT to 15:50 UT, 11 February (corresponding to the blue dashed line in Figure 1). Green diamonds show the ACE differential intensity integrated over 09:59 UT to 12:00 UT, 11 February (corresponding to the green dashed line in Figure 1).

Because the coronal structures from which fast solar wind originated from often evolve slowly, CIRs are considered as a steady state structure in the corotating frame to the first order [Mason et al., 2009]. One therefore expects to observe the same reverse shock at ACE after ∼1.6 days. Indeed, the He intensity observed by ACE began to increase gradually and reached its maximum at 22:30 UT, 10 February. However, the proton speed, magnetic field, and total pressure showed no discontinuities, so no reverse shock was observed by ACE. This is not surprising since time variations of plasma properties such as density and speed can cause the location of the shock to vary. The energy spectrum from ACE observations for the period between 09:59 UT and 12:00 UT, 11 February, is shown as the green curve in Figure 2. During this time, ACE is magnetically connected to the shock at a location not far from 1 AU. Consequently, there is little modulation at low energies.

2.1. Event Geometry Figure 3 shows the relative locations of three spacecraft at 20:00 UT, 9 February 2008, just before the energetic He intensities began to increase on STB. STA, STB, and Earth are represented by red (labeled as A), blue (labeled as B), and green solid circles, respectively. The separation angle between STB and Earth (ACE) is 23.656∘ , and the heliocentric distance of STB is 1.0 AU and that of Earth is 0.987 AU. (http://stereo-ssc.nascom. nasa.gov/cgi-bin/make_where_gif) Since ACE lies at the L1 Lagrange point between the Sun and the Earth, its heliocentric distance is 0.977 AU. The configuration of magnetic field lines as well as the CIR reverse shock with respect to STB and ACE is shown in the cartoon in Figure 4. The Sun is located at O. The black half circle is the trajectory of STB, and the dashed half circle is the trajectory of ACE in the corotating frame. Green, blue, and red curves are Parker field lines. The black thick curve extending from A to D represents the CIR reverse shock surface. Point A (B) is the intersection between the blue (red) magnetic field line and the STB orbit. Point H (I) is the intersection between the green (blue) magnetic field line and the ACE orbit. When STB is at location A, it connects to the local reverse shock, corresponding to observations at 19:36 UT, 9 February 2008. As STB rotates from A to B in the corotating frame, its connection point at the shock (i.e., the acceleration site) moves farther out. When STB is at location B, under the assumption of no cross-field diffusion, it will observe energetic particles that are accelerated by the reverse shock at point D. Figure 3. Relative locations of STA and STB and Earth at 20:00 on 9 February 2008. STA and STB are indicated by red A and blue B. Earth and Sun are indicated by green and yellow solid circles. Solid curves are Parker’s magnetic field lines. Separation angle between STA and Earth is 23.656∘ . Separation angle between STA and Earth is 21.833∘ . Heliocentric distance of STA is 0.966 AU and that of STB is ∼1.0 AU.

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As shown in Figure 1, the intensity of STB He gradually decreases from 19:36 UT, 9 February, to 13:55 UT, 11 February. This decrease is due to two reasons. First, as the acceleration sites moves out, it becomes harder and harder for accelerated particles to propagate back to 1 AU. This is the modulation effect. Second,


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the density of seed particles; therefore, the intensity of the accelerated particles may also vary with heliocentric distance. Note that the composition of the seed particle is still presently under debate. In modeling SEP events, where ions and electrons are accelerated at a CME-driven shock that propagates out from the Sun, it has been argued and assumed that a fraction of the solar wind (e.g., 1%–4% as often used in Li et al. [2003] and Li and Zank [2005]) is accelerated at the CME-driven shock. CIR shocks, however, are often quasi-perpendicular shocks and have higher injection energies than quasi-parallel CME-driven shocks. So it is hard for bulk solar wind to be accelerated. Recently, Mason et al. [2012b], through a study of the abundance of 3 He and He+ , have suggested that the seed particles for CIR shocks is the suprathermal ion pool rather than the bulk solar wind. If the seed particle was the solar wind, Figure 4. Configuration of CIR with respect to STB and ACE. The Sun is then the density of the seed particles will located at O, black half circle is trajectory of STB in corotating frame, have an r−2 dependence. However, if the and dashed half circle is trajectory of ACE. Green, blue, and red curves are Parker’s spiral magnetic field lines. A(B) is the intersect point of blue seed particle was the suprathermal ions, then the radial dependence of the seed (red) curve and STB’s orbit. H (I) is the intersect of green (blue) curve and ACE orbit. Black thick curve (dashed thick curve) extending from A particle may be more complicated. For to D is reverse shock surface. example, the cooling of solar wind may imply faster decay of the seed particles than r−2 . On the other hand, some continuous particle-wave interaction in the solar wind may suggest a slower decay than r−2 . Indeed, using Cassini observations, Hill et al. [2009] examined how the intensities of 2–60 keV/nucleon suprathermal He++ vary with r. They found that the intensities of suprathermal He++ decrease slower than r−2 [see Hill et al., 2009, Figure 2]. In our event, because STB observed the shock in situ at 1 AU, if we assume the shock is a steady structure in the corotating frame and if shock parameters do not change significantly along the shock surface, then with a particle transport model, we can examine how the intensity of the accelerated particles at the shock varies with radial distance. From in situ observation of STB, we find that the average solar wind speed in the fast compression region between 12:00 and 18:00 UT, 9 February, is vcomp = 620 km s−1 and in the fast solar wind right after the shock passage is vsw = 760 km s−1 (see Figure 1). Correspondingly, the angle between the upstream magnetic field with the radial direction is ∼30∘ . The reverse shock has a 𝜃BN = 51° (see the online list http://www-ssc.igpp. ucla.edu/∼jlan/STEREO/Level3/STEREO_Level3_Shock.pdf), so the angle between the shock normal and the radial direction, 𝛼 , is 20∘ . If we approximate the shock normal to be along the radial direction, then we can estimate the shock speed vsh at 1 AU from vsh =

𝛽vcomp − vsw 𝛽−1

.

(1)

Here 𝛽 = 2.17 is the compression ratio of the reverse shock as obtained from in situ magnetic field data. The calculated shock speed is 500 km s−1 in the s/c frame. Note that vsh < vsw since the reverse shock propagates toward the Sun in the fast solar wind frame. Assuming vsh does not vary with r, we can obtain the radial distance OD from, vsw CE = . (2) DE vsw − vsh ZHAO ET AL.


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Using equation (2), with the above vsw and vsh , we find CD CE − DE = = 1.92, DE DE

(3)

DE = vsw Δt,

(4)

and the length of DE can be calculated by

where Δt = 1.84 days is the time difference between points A and B. So the distance between C and D is 1.55 AU. Therefore, for STB observation at point B, the energetic particles are accelerated at point D on the reverse shock, having a heliocentric distance rsh = 2.55 AU. We remark that equation (1) is only an approximation and it is only valid when 𝛼 is small (note that cos 20° = 0.94). If the shock normal differs from the radial direction substantially (e.g. 𝛼 > 30°), one cannot use equation (1). Now consider the ACE observations. Given that the sidereal rotation period of the Sun is 24.47 days and the angle between STB and ACE is ∼23.656∘ , it takes 1.61 days for ACE to be connected to the exact same portion of the shock structure as STB, implying the enhancement of particle intensity in ACE will be delayed by 1.61 days from STB. From the observations shown in Figure 1, we find a time delay of the peak intensity between STB and ACE to be 1.14 days. This difference of d𝜏 between the observation and by considering solar rotation can be due to a number of reasons. For example, the ACE and the STB have different latitudes, so ACE and STB did not see the same portion of the shock. Finally, the CIR shock may not be completely stationary and its location may vary with time (similar to, e.g., the heliospheric termination shock). The effect of this time variation is illustrated in Figure 4. In the cartoon, the dashed thick curve depicts the CIR reverse shock when ACE is at point H. Since ACE did not observe the shock in situ, it is hard to discern the exact cause of d𝜏 . In the following, we do not consider possible oscillations of the CIR shock and assume that it is given by the thick solid curve as in Figure 4. Following the analysis as in the STB case, we find that for a later time of ACE observation at 10:00 UT 11 February, the source location at the shock has a rsh of 1.39 AU. We perform two numerical simulations for two 2 h periods corresponding to STB observations at 13:55–15:55 UT 11 February and ACE at 10:00–12:00 UT 11 February. The periods are chosen in which the energetic particle intensities have small variations. We note that STA also observed this event about 2 days later than ACE. However, the intensity profile from STA observation showed clear differences from STB and ACE. This can be due to the fact that assumptions of a steady state CIR reverse shock is only applicable within a short duration. As shown in Mason et al. [2009, Figure 6], energetic particle observations of the same CIR event with a few days apart can differ substantially. In our event, the separation between STA and STB is ∼45∘ , >3 days apart, and we do not consider STA observations in this study. As noted before, we assume that the shock is in a steady state in the corotating frame and that the shock strength does not vary along the shock surface. Assuming the shock parameters do not vary as a function of heliocentric distance allows us to use the 1 AU in situ shock spectrum from STB as a reference for energetic particle spectra at different times. Therefore, as noted before, we can decouple the transport process from the acceleration process for CIR-associated energetic particles. Specifically, we will use the in situ STB 1 AU shock observation of the energetic particle differential current intensity J(E) and scale it by a factor of 𝜂 (defined below) at two different locations at the CIR reverse shock, to calculate the resulting differential intensities as observed at 1 AU by STB and ACE. By comparing simulations with observations, we can obtain how 𝜂 , which is a measure of the seed particle intensity, varies with r. The implication of this radial dependence of 𝜂 is discussed in section 4.

3. Model Description We use a Monte Carlo code to study the transport of charged particles. The transport of energetic particles can be described by the focused transport equation [e.g., Skilling, 1971; Isenberg, 1997]: ( [ )] 𝜕u 1 − 𝜇 2 𝜕uj 𝜇bi 𝜕𝜇 𝜕ui 1 − 3𝜇 2 𝜕f 𝜕f 𝜕f + (ui + v𝜇bi ) bi bj i − + + − v 𝜕t 𝜕xi 2 𝜕xj 2 𝜕xj v 𝜕t 𝜕xj 𝜕v (5) [ ( )] 𝜕uj 2bi 𝜕u 𝜕ui 𝜕ui 1 − 𝜇 2 𝜕bi 𝜕f 𝜕f 𝜕 + v + uj = D + S − L. +𝜇 − 3𝜇bi bj − 2 𝜕xi 𝜕xi 𝜕xi v 𝜕t 𝜕xj 𝜕𝜇 𝜕𝜇 𝜇𝜇 𝜕𝜇 ZHAO ET AL.


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In the above, the distribution function f is assumed to be gyrotropic; the bi and ui are the components of the unit magnetic field and solar wind speed. The particle’s pitch angle cosine is 𝜇 . S and L are the source and loss terms. The focused transport equation can be derived from the guiding center theory [le Roux and Webb, 2009]. The particle’s motion therefore can be regarded as the motion of the guiding center along the background magnetic field line plus a diffusion in pitch angle, governed by D𝜇𝜇 . In the solar wind, the background magnetic field is given by R0 2 ) , r B𝜃 = 0, and Ωr sin 𝜃 (r > > R0 ), B𝜙 = Br u Br = B0 (

(6)

where 𝜃 = 90° corresponds to the magnetic field in the ecliptic plane. In this work, we use B0 = 1.53 Gauss at R0 = 1 Rs (solar radius). Energetic particles are followed in two different frames [see Ruffolo, 1995; Agueda et al., 2005; Kocharov et al., 2003]. One is the instantaneous corotating frame; the other is the instantaneous solar wind frame. The instantaneous corotating frame is a frame which corotates with the Sun. The instantaneous solar wind frame is a local inertial frame that comoves with the solar wind at the particle’s location. In the instantaneous corotating co = vsw ∕ cos 𝜓 , where 𝜓 is the angle between the magnetic field direcframe, the solar wind velocity is vsh tion and the radial direction. The solar wind velocity is then parallel to the local magnetic field, yielding a zero-induced electric field. Therefore, the particle’s energy is conserved in this frame. Furthermore, due to the focusing effect, the particle’s pitch angle will change because of the conservation of the particle’s magnetic moment p2⟂ ∕B. At each time step, the particle’s location and pitch angle are updated in this instantaneous corotating frame. We then transform the particle’s momentum to the instantaneous solar wind frame. This is the frame where the effect of solar wind magnetohydrodynamic (MHD) turbulence is considered since MHD waves and turbulence are generated locally. Pitch angle diffusion is considered in this frame. After considering pitch angle diffusion, we transform back to the instantaneous corotating frame. Finally, we need to transform the particle’s energy and momentum between two instantaneous corotating frames at two different times t and t + 𝛿t. As shown in Appendix A, to the order of (v∕c)2 , the particle’s energy is conserved under the Lorentz transformation between these two frames. In modeling the pitch angle diffusion, we follow Zhao and Li [2014]. For each time step 𝛿t, the particle’s pitch angle 𝜇 will change by a small amount 𝛿𝜇 given by √ ) ( 𝜕D𝜇𝜇 1 𝛿t; 𝛿𝜇 = sign 𝜖 ′ − erf−1 (𝜖) 4D𝜇𝜇 𝛿t + 2 𝜕t

(7)

here 𝜖 ′ and 𝜖 are random numbers uniformly distributed in [0, 1]. The inverse of error function is erf−1 . In quasi-linear theory (QLT) [Jokipii, 1971; Lee and Lerche, 1974; Wentzel, 1974], D𝜇𝜇 is given by D𝜇𝜇 =

kP(k) 𝜋 1 < (𝛿𝜇)2 > = (1 − 𝜇 2 )Ωo 2 , 2 𝛿t 4 Bo

(8)

where P(k) (given in equation (9)) is the turbulence power spectrum in solar wind. Ωo = eB∕(𝛾m) is particle’s gyrofrequency. Particle’s resonant wave number is k = Ωo (v|𝜇|)−1 . P(k) = Aq 𝜆c (𝛿B)2 (1 + (k𝜆c )q )−1 .

(9)

𝜆c is the correlation length and q is the power law index of the turbulence spectrum. A𝛽 is determined by the normalization condition k L

∫ks

P(k)dk = (𝛿B)2 .

(10)

The correlation length 𝜆c , smallest (kS ), and largest (kL ) wave number in P(k) are set to be the typical values at 1 AU [see Zhao and Li, 2014]. ZHAO ET AL.


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The radial dependence of B is well defined for a Parker field. For 𝛿B, however, the radial dependence is still presently unknown. Some earlier studies by Bruno and Carbone [2013] suggested that a WKB approximation, i.e., 𝛿B2 ∼r−3 , may be reasonable but slightly underestimated. Following Mason et al. [2012a], we use 𝛿B2 ∼r−3.5 in this work, so 𝛿B2 (r) = 𝛿B2 (1AU)

Figure 5. Injection energy spectrum f (E) obtained from Figure 2. f (E) = J(E)∕v , where v is particle’s speed. f (E) is composed of two parts. The first part is E −3.57 between 0.094 MeV/nucleon < E < 0.546 MeV/nucleon, and the second part is E −4.89 between 0.546 MeV/nucleon < E < 2.185 MeV/nucleon.

(

r 1AU

)−3.5

.

(11)

We assume that the shape of the source energetic particle spectrum f (E) ∼ J(E)∕v does not vary with r and is given by a broken power law form as shown in Figure 5. The form of f (E) is chosen to fit the observed differential intensity J(E) at STB immediately downstream of the shock between 17:54 and 20:09 UT, 9 February. In Figure 5, the low-energy portion of f (E) (shown in blue) has f (E) ∼ E −3.57 between 0.094 < E∕(MeV/nucleon) < 0.546, and the high-energy portion (shown in red) has f (E) ∼E −4.89 between 0.546 < E∕(MeV/nucleon) < 2.185.

For our simulation, we inject a total of N0 = 60, 000 protons at location of rsh (which is also the outer boundary) with an initial pitch angle cosine 𝜇 uniformly distributed between −1 and 0. We do not consider the acceleration process in this work. So these particles are injected with a given spectrum. We assume this spectrum, up to a factor, is the same as that from the in situ observation of the shock at STB. In this way, we attempt to decouple the acceleration process from the transport process. Note that close to the shock, the acceleration process is governed by the diffusion coefficient, which can be substantially smaller than that in the interplanetary medium. So the acceleration process is quite different from the transport process. We release all particles at the same time and follow all of them for a period of 10 days and obtain the time-integrated spectrum. We then follow the transport of these particles from the shock to 1 AU. Particles leave the simulation domain when they reach either the inner boundary (r = 0.01 AU) or the outer boundary (r = rsh ). The particles’ momenta and pitch angles are recorded when they pass r = 1 AU.

4. Results and Discussion The simulated proton differential intensities are shown in Figure 6 as the black curves with “diamond” symbols for two cases. Figure 6 (left) is for case I, corresponding to the STB observation between 13:55 UT and 15:50 UT, 11 February. In this case, energetic protons are injected at a heliocentric distance r=2.55 AU, and the particle differential current is obtained at 1.0 AU. The observed differential current intensity is shown as the blue curve with the “plus” sign. The right panel is for case II. In this case, energetic protons are injected at a heliocentric distance r = 1.39 AU and the particle differential intensity is obtained at 0.98 AU. This case corresponds to the ACE observation between 10 ∶ 00 UT ∼12:00 UT, February 11. The observed differential intensity is shown as the green curve with the plus sign. In both cases, we use a 2 h observation window so that enough statistics can be obtained. We assume that the shock location does not change during this 2 h interval. We do not consider a longer period (>2 h) because the acceleration site to which the spacecraft is connected to can change rapidly along the shock surface. To fit the observed differential intensities at 1 AU in Figure 6, a key parameter is (𝛿B∕B)2 at 1 AU. We calculate (𝛿B∕B)2 using the STB/MAG 1 min data and the ACE/MAG 4 min data. In the case of STB observation, the average magnetic field was obtained for a period of 20 h before the 2 h observation window, and we find (𝛿B∕B)2 = 0.010. For the ACE observation, the background magnetic field before the 2 h observation window showed a clear decreasing trend, so we have chosen a 20 h period after the 2 h observation window to calculate the average magnetic field and (𝛿B∕B)2 . We find (𝛿B∕B)2 = 0.019 in this case. The magnetic field data and the periods for calculating the average background magnetic field and (𝛿B∕B)2 for both the ACE and STB observations are shown in Figure 7. Figure 7 (left column) is for STB, and Figure 7 (right column) is for ACE. In both cases, Figure 7 (top row) shows the total B for an extended period covering the whole event, and Figure 7 ZHAO ET AL.


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Figure 6. Comparison of CIR-associated energetic particle differential intensities between observations and simulations. (left) For STB observation (case I) and (right) for ACE observation (case II). The turbulence level at 1 AU for case I is 𝛿 2 B∕B20 = 0.012, and for case II is 𝛿 2 B∕B20 = 0.02. Simulation results are shown in black, and observations are shown in green and blue.

(bottom row) is zoom-in plots of the total magnetic field and (𝛿B∕B)2 for a 20 h period before (for the case of STB) or after (for the case of ACE) the 2 h observation window of energetic particles. Now consider the fitting of STB observation in Figure 6. We vary the ratio of (𝛿B∕B)2 at 1 AU from the observed in situ value of 0.01 to fit the shape of the observed differential intensity at 1 AU. Note that the observed (𝛿B∕B)2 = 0.01 is an ensemble average of many radially propagating plasma parcels that pass through the spacecraft during a 20 h window. These plasma parcels do not consist of a Parker field line on which energetic particles propagate from 2.55 AU to 1 AU. So it is only a proxy of the turbulence level along the Parker field line particles propagate on. Nevertheless, the best fit yields a (𝛿B∕B)2 = 0.012 at 1 AU, very close to the in situ observation. The simulated and observed differential current intensities for STB observation agree nicely as can be seen from Figure 6 (left). Next consider the ACE observation. Again, by varying the ratio of (𝛿B∕B)2

Figure 7. (left column) STB and (right column) ACE observations of the magnetic field. (top left) Observation of the magnetic field for the STB observation. (top right) Observation of the magnetic field for the ACE observation. (bottom left) A 20 h zoom-in plot of Figure 7 (top left). Red curve is (𝛿B∕B)2 . (bottom right) A 20 h zoom-in plot of Figure 7 (top right). Red curve shows (𝛿B∕B)2 .

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Table 1. Monte Carlo Results Shock Location

(𝛿B∕B)2

Spacecraft

𝛼

𝜂

Case I

2.55 AU

0.012

STB

6

1.27

Case II

1.39 AU

0.02

ACE

1

0.61

at 1 AU, the best fit yields (𝛿B∕B)2 = 0.02, close to the in situ observation of (𝛿B∕B)2 = 0.019 at ACE. This is shown in Figure 6 (right). After fitting the spectral shape, we next obtain the normalization factor 𝜂 of the injected particles from the fitting. The parameter 𝜂 is defined through E

𝜂=

∫E max dEJsh min

E

∫E max dEJ1AU

,

(12)

min

where Jsh is the differential intensity at the shock (r = 1.39 AU for the ACE observation and r = 2.55 AU for the STB observation) and J1AU is the differential current when the shock is observed in situ at 1 AU by STB between 17:54 and 20:09 UT, 9 February. The parameter 𝜂 reflects how the number of seed particles, i.e., the particles that participated in the shock acceleration process, depends on r. In the simulation, we inject particles to a flux tube whose cross section depends on the shock location. Because the foot point of the flux tube on the shock surface varies with heliocentric distance, we introduce an additional parameter 𝛼 through 𝛼=𝜂

Ash , A1AU

(13)

where Ash is the cross section of the flux tube that intersects with the shock at r, and A1 AU corresponds to the in situ STB observation of the shock at 1 AU. The factor Ash ∕A1AU reflects how the flux tube expands as a function of r. Knowing the solar wind speed and the shock speed, the ratio Ash ∕A1AU can be readily calculated as shown in Appendix B. For our cases, A1.39AU ∕A1AU = 1.65, and A2.55AU ∕A1AU = 4.72. From the fitting we obtain 𝛼 = 6 for case I and 𝛼 = 1 for case II. Therefore, we obtain 𝜂case I = 1.27 and 𝜂case II = 0.61. Our simulation results are summarized in Table 1. These values of 𝜂 ’s are very important results. Consider first the ACE observation with rsh = 1.39AU. In this case we find that to fit the observation, the number of accelerated particles at rsh is 0.6 times that at 1 AU. This decrease yields a radial dependence of r−1.55 . This is shallower than the radial dependence of solar wind density r−2 and is consistent with Hill et al. [2009] where the intensities of suprathermal He++ was found to decrease slower than r−2 . Therefore, the ACE observation supports the notation that the seed particles for CIR-associated energetic particles are most likely suprathermal ions than the bulk solar wind [Mason et al., 2012b]. Comparing to the ACE observation, the STB observation with rsh = 2.55 AU shows that Jsh (r = 2.55AU)∕Jsh (r = 1AU)=1.27. So the number density of the accelerated particles at 2.55 AU has to be larger than that at 1 AU. This implies that the intensity of the seed particles increases with r, instead of decreasing with r. It contradicts with the results in Hill et al. [2009]. However, we note that the data points in Hill et al. [2009] are 0.5–1 AU apart, and most are far beyond the Earth orbit, so they may not provide much constraint about the relatively small shock distances studied here. A larger seed population further out than 1 AU may seem counterintuitive. However, earlier works [McDonald et al., 1976; van Hollebeke et al., 1978] have shown that CIR intensities increase beyond 1 AU and peak at several AU. Note that in these earlier studies, the enhancement of CIR intensity is largely attributed to the fact that CIR shocks tend to form beyond 1 AU [e.g., Desai et al., 1999]. In our case, however, the CIR shock was seen to form in situ at 1 AU by STB. Note that we have assumed that the shock parameters do not vary with the heliocentric distance of the shock. It is possible that the shock strengthens beyond 1 AU in that the injection energy decreases with r, so that more particles can participate in the shock acceleration process. This, of course, still implies that the seed population (i.e., particles participating the shock acceleration process) increases with r. This increase of seed particle intensity at 2.55 AU is required because the modulation effect for low-energy particles is significant. This can be seen from the following arguments. Under the assumption that particles do not diffuse across the field lines, the path length from 2.55 (1.39) AU to 1 AU is 2.26 (0.52) AU (assuming a solar ZHAO ET AL.


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Figure 8. The Fisk-Lee fitting of the differential intensities at (1) (left) the reverse shock as observed in situ by STB during 17:54 UT to 20:09 UT, 9 February 2008, (2) (middle) upstream of the reverse shock for ACE observation during 02:30 UT to 04:30 UT, 11 February 2008, and (3) (right) upstream of the reverse shock for STB observation during 00:00 UT to 02:00 UT, 11 February 2008.

wind speed of 760 km s−1 ). If there is no scattering (𝛿B∕B0 → 0), then as a charged particle moving toward the Sun along the Parker field, the focusing effect (conservation of particle’s magnetic moment) will tend to reverse the particle’s momentum direction. With turbulence included, the pitch angle will undergo both a focusing and a scattering process. To compete with the focusing effect, one may think that a larger 𝛿B∕B0 will help. However, if 𝛿B∕B0 is too large so that pitch angle scattering dominates focusing, then the motion of particles along the field can be regarded as a diffusion and the time to arrive 1 AU from the shock will be very long. For example, if the mean free path of a 0.5 MeV proton is ∼0.5 AU [see, e.g., Li et al., 2003, equation (27)], it takes ∼10 mean free paths for an 0.5 MeV proton to arrive 1 AU from 2.55 AU, translating to a time period of 21 h. Since the adiabatic cooling rate is ∇ ⋅ vsw , the longer the propagation time, the more deceleration a particle will experience, leading to a stronger modulation. If the mean free path (mfp) 𝜆 is smaller, since the travel time scales as 𝜆−1 , the adiabatic cooling will lead to a stronger modulation. Finally, for particles of smaller energies, using equation (27) of Li et al. [2003] (with 𝛼 = 1∕3), one finds that the travel time scales roughly as E −2∕3 , so the modulation will be even stronger. The above discussion illustrates why a significant number of seed particles has to be present when the shock is further out. Note that one implicit assumption in our scenario is that particles are tied to a single magnetic field line. So we have ignored 𝜅⟂ . The picture can be changed, and perhaps substantially, if we allow particles to diffuse across field lines. This can be seen easily from Figure 4. Within a small longitudinal range we see that the shock goes from 1 AU (point A) to 2.55 AU (point D). While it is difficult for low-energy particles accelerated at point D (2.55 AU) to propagate along the Parker spiral to point B (1 AU), if the low-energy particles could cross-field diffuse from field lines that connect to the shock at a closer distance than 2.55 AU (e.g., field lines to the left of BD), then their propagation distance would be much smaller . Consequently, if cross-field diffusion plays an important role, then we do not need to require as large a seed population further out. We note that if particles could cross-diffuse into, e.g., the field line BD as shown in Figure 4, particles on the field line of BD could also cross-diffuse out. However, since low-energy particles accelerated at point D will have a hard time to propagate to point B, so whether they stay in the same field line or diffuse out to other field lines does not matter much to observations at point B. It is instructive to fit the differential intensities for both case I and case II using the Fisk and Lee model. Following Fisk and Lee [1980], one can show that the upstream differential intensity J(E) is ( )2∕(𝛽−1)+V∕(𝜅0 v) R J = J0 v n e−v∕v0 , (14) Rs (

where 𝛽=

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2 Vup + Ω2 R2s 2 Vdn + Ω2 R2s


)

Bdn . Bup

(15)

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Table 2. Fisk and Lee Fitting Shock Location

𝜅0

Spacecraft

J0 (# cm−2 s−2 sr−1 (MeV/nucleon)−1 )

𝜂

Source

1.0 AU

0.03

STB

1227

1.0

Case I

3.63 AU

0.03

STB

5474

3.4

Case II

1.73 AU

0.03

ACE

4167

4.7

2

In the above 𝛽 is the compression ratio, n = −(𝛽 + 2)∕(𝛽 − 1); v0 = V(𝛽−1) ; V the solar wind speed; Ω the 6𝜅0 𝛽 angular rotation speed of the Sun; subscripts “up” and “dn” refer to quantities upstream and downstream of the CIR reverse shock; and J0 is a fitting constant. At the CIR shock, J = J0 v n e−v∕v0 . There are four free parameters in equation (14). These are Rs , J0 , 𝛽 , and v0 . In fitting the observed CIR-associated energetic particle spectrum using the functional form of equation (14), one issue is that for rather different choices of the parameter set (J0 , 𝛽 , and v0 ), one can obtain very comparable fitting results. If, however, we assume that the solar wind speed and the shock compression ratio do not vary as a function of heliocentric distance, then 𝛽 is fixed by the STB in situ observation. Furthermore, v0 is also fixed by the STB in situ observation. Therefore, the only two free parameters are J0 and Rs . Figure 8 shows the fitting results for the differential intensities for (1) STB in situ observation of the reverse shock during 17:54 UT to 20:09 UT, 9 February 2008, (2) ACE observation during 02:30 UT to 04:30 UT, 11 February 2008, and (3) STB observation during 0:00 UT to 02:00 UT, 11 February. To obtain the best fitting, these are slightly different time periods from our simulation. Fitting the differential intensity at the reverse shock as observed by STB using a compression ratio 𝛽 = 2.13 from the in situ measurement, we find J0 (in unit of cm−2 s−2 sr−1 (MeV/nucleon)−1 ) is 1227 and v0 = 0.172 (MeV/nucleon)1∕2 (corresponding to 𝜅0 = 0.03). Using these two parameters, we then find J0 (ACE)= 4167 and Rs = 1.73 AU; J0 (STB)= 5474 and Rs = 3.63 AU. The fitting results of J0 and Rs are summarized in Table 2. These shock locations are to be compared with our simulations where Rs = 1.39 AU for the ACE observation and Rs = 2.55 AU for the STB observation. The arbitrariness of the fitting can be seen from the following: for the ACE fitting, for example, we can obtain an almost equally well fitting with J0 (ACE)= 2070 and Rs = 1.5 AU. In any events, we see that the Fisk and Lee model yields somewhat larger shock locations for both periods. Now considering the differential intensities: the Fisk and Lee model predicts a differential intensity at Rs = 1.73 AU = 3.4 times that when Rs = 1 AU (or a differential intensity at Rs = 1.5 AU to be 2070 = 1.69 times to be 4167 1227 1227 5474 that when Rs = 1 AU); and a differential intensity at Rs = 3.63 AU to be 5160 = 4.7 times that when Rs = 1 AU. Therefore, the Fisk and Lee model also suggests that the differential intensity at the shock increases with heliocentric distance of the shock.

5. Conclusion We have examined energetic particle transport in one CIR event which occurred in February 2008. We choose this event because the CIR was observed by both STB and ACE and the CIR-associated energetic particle intensity profiles from these two spacecraft reasonably resemble each other (with a time shift of ∼1.4 days). Furthermore, STB observed the reverse shock in situ, suggesting that the shock was formed near 1 AU. Under the assumption of no cross-field diffusion, we develop a Monte Carlo test particle model to investigate the transport of energetic particles. The model solves the focused transport equation numerically. Comparing to previous analytical work by Fisk and Lee [1980], our model considers explicitly the particle pitch angle evolution. Both the magnetic focusing effect and the pitch angle diffusion process are included. For the cases where the solar wind MHD turbulence is not strong, our approach is more appropriate than that of Fisk and Lee [1980]. Assuming the reverse shock can be approximated by a stationary structure in the corotating frame, and that the shape of the accelerated particle spectrum at the shock does not vary with heliocentric distance, we calculated the different intensity at 1 AU for two periods corresponding to an ACE observation and a STB observation. By assuming the accelerated particle spectrum at the shock is given by that observed in situ at STB and does not vary with heliocentric distance, we avoid considering particle acceleration at the shock explicitly. Presumably, the acceleration at the shock does vary with the heliocentric distance. Under the diffusive shock acceleration framework, the accelerated particle spectrum depends on various shock parameters ZHAO ET AL.


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including the shock geometry and the shock compression ratio. Both can depend on the heliocentric distance. However, since we have only 1 AU observations, no constraints on these parameters can be obtained. We note this as a limitation of the present work. By assuming a turbulence similar to that given by the WKB approximation, reasonable agreements between the simulation and the observations are obtained for both observations. The best fit of the STB observation yields (𝛿B∕B0 )2 = 0.012, similar to the in situ value of 0.01. The best fit of the ACE observation yields (𝛿B∕B0 )2 = 0.02, also similar to the in situ value of 0.019. The ACE fitting suggests that the seed particle density at rsh = 1.39 AU is 0.61 times that when rsh =1 AU, consistent with the radial dependence of suprathermal He++ obtained in Hill et al. [2009]. However, the STB fitting suggests that the seed particle density at rsh = 2.55 AU has to be 1.27 times larger than that when rsh = 1 AU. This contradicts to Hill et al. [2009]. This requirement of a large seed particle density at rsh = 2.55 AU is due to the fact that the modulation, especially for low-energy particles, is strong. This contradiction may be resolved by including cross-field diffusion. Including cross-field diffusion can effectively negate the modulation effect since with cross-field diffusion included, low-energy particles that accelerated at a shock location closer to 1 AU can diffuse cross field and contribute to the observed intensity at a location that magnetically connects to the shock at a large distance (e.g., ∼2.55 AU). In summary, we have developed a numerical Monte Carlo code to examine energetic particle transport at CIR shocks. We ignored particle cross-field diffusion and consider particles propagating along a single field line. We examined one CIR event which occurred in February 2008 using our code. We simulated energetic particle spectra for two 2 h windows at both ACE and STB. Very good agreement between the simulation and observation can be obtained if we assume the seed particles at 2.55 AU is ∼1.27 times that at 1 AU and if the seed particles at 1.39 AU is ∼0.61 times that at 1 AU. However, we note that this conclusion may be changed if cross-field diffusion is included.

Appendix A: Instantaneous Corotating Frame Transformation The adiabatic deceleration effect in an expanding solar wind is described by Parker [1965] and Jokipii and Parker [1970] p < ṗ > = − ∇ ⋅ vsw , 3

(A1)

where vsw is the solar wind speed and p is particle’s momentum in solar wind frame. This expression of average deceleration rate is valid under the assumption that particle’s pitch angle diffusion is nearly isotropic [Ruffolo, 1995]. In Ruffolo [1995] the analytical expressions of the particle’s momentum (pitch angle) deceleration rate for an individual particle is given by [

] sec𝜓 d 2 2 (1 − 𝜇 ) + cos 𝜓 sec𝜓𝜇 ṗ = −pvsw 2L(z) dr ] ) ( [ vsw v vsw d v sec𝜓 − 𝜇 2 sec𝜓 (1 − 𝜇 2 ) − vsw cos𝜓 sec𝜓 𝜇(1 − 𝜇 2 ), 𝜇̇ = 1+𝜇 2L(z) v dr c

(A2)

where 𝜇 denotes the particle’s pitch angle in solar wind frame. The angle between the radial direction and the magnetic field’s tangent direction is 𝜓 . 1∕L(z) = −B(z)∕(𝜕B∕𝜕z) is the reciprocal of the scale length of the interplanetary magnetic field. The first equation in equation (A2) describes the adiabatic cooling effect, and the second equation describes pitch angle focusing effect. In this paper, particle’s adiabatic deceleration and focusing effect are not modeled by equation (A2), instead these effects are treated implicitly by a frame transformation approach. Figure A1 shows the local coordinates of two adjacent instantaneous corotating frames. In Figure A1a, the black curve is the spiral magnetic field line, and A, B are two adjacent locations with a time interval dt. The radial distance of A is R1 and of B is R2 . The unit vector in the direction of R1 (R2 ) is r1 (r2 ). The unit vector in the direction perpendicular to r1 (r2 ) is 𝜽1 (𝜽2 ), which is also the direction of instantaneous corotating frames at location A (B). The four vectors’ relative directions are shown in Figure A1b. Δ𝜓 is the angle between r1 and r2 . ZHAO ET AL.


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Suppose the Sun rotates with an angular velocity Ω, we get the velocities of the two instantaneous corotating frames V 1 = ΩR1 𝜽1 , V 2 = ΩR2 𝜽2 ,

(A3)

and the relative velocity between two corotating frames is V 1 − V 2 = ΩR1 𝜽1 − ΩR2 𝜽2 .

(A4)

Take the dot product of (V 1 − V 2 ) and AB and substitute equation (A3), we get (V 1 − V 2 ) ⋅ (R1 𝜽1 − R2 𝜽2 ) = ΩR21 𝜽1 ⋅ r1 + ΩR22 𝜽2 ⋅ r2 − ΩR1 R2 𝜽1 ⋅ r2 − ΩR2 R1 𝜽2 ⋅ r1 . Figure A1. Sketch of frame transformation. (a) A and B are two adjacent points in a Parker spiral. The radial distance of A is R1 and of B is R2 . The unit vector in the direction of R1 (R2 ) is r1 (r2 ). The unit vector in the direction perpendicular to r1 (r2 ) is 𝜽1 (𝜽2 ). Δ𝜓 is the angle between r1 and r2 . (b) Relative directions of the four unit vectors.

(A5)

Clearly from Figure A1b, r1 is perpendicular to 𝜽1 and r2 is perpendicular to 𝜽2 , then

r1 ⋅ 𝜽1 = 0, r2 ⋅ 𝜽2 = 0, r1 ⋅ 𝜽2 = sin(Δ𝜓),

(A6)

r2 ⋅ 𝜽1 = − sin(Δ𝜓).

Combining equations (A5) and (A6), the dot product of (V 1 −V 2 ) and AB is zero, which means if the time step is small, the relative velocity between two corotating frames is perpendicular to the Parker’s spiral (particle’s trajectory). One can then obtain the particle’s energy and the momentum component parallel and perpendicular to the Parker’s spiral from the Lorentz transformation.

Appendix B: Determination of Shock Cross Section Total number of injected particles (N0 ) at CIR reverse shock plays an important role in the simulationobservation comparison. Assuming the shock strength does vary along shock surface, the total number of injected particles is proportional to the seed particle density (n) and the shock cross section (A) in a flux tube. Figure B1 shows the configuration of the reverse shock. Similar to Figure 4, red and blue curves are Parker spirals, black thick curve extending from A to D is the reverse shock surface. The Sun is located at O. Red (blue) Parker spiral intersects the radial direction at A and E (B and D). The length OA is the same as OB and OC , which is r. The vectors n, r, r′ , and t are unit vectors in the direction of shock normal, OA, OB, and tangent direction of the Parker spiral colored in red at point A, respectively. The following statements and calculations are based on the approximation that 𝜃 , which is the angle between OA and OB, is small. Then, AB is perpendicular to OA and OB; BC is perpendicular to OB and OD; AB and BC are in the direction perpendicular to the radial direction. Angle between r and t is 𝜓 , and we have tan𝜓 = Bt ∕Br (Bt and Br are given in equation (6)). The angle between r′ and BD is 𝛾 , and 𝛽 is the angle between BD and CD. And in the approximation, we have 𝛾 = 𝜓 and 𝛽 = 𝜓 . Angle between shock surface and AB is 𝛼 , and the distance of CD is dr. Follow the law of cosine, in the triangle ABD, we get AD2 + AB2 − 2ADAB cos 𝛼 = BD2 .

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Figure B1. CIR reverse shock configuration. The Sun is located at O. Red and blue curves are Parker’s spiral. Black thick curve extending from A to D is reverse shock surface.

Acknowledgments We thank the many individuals at the University of Maryland, Johns Hopkins Applied Physics Laboratory, and Goddard Space Flight Center responsible for the construction of the ACE/ULEIS and STEREO/ IMPACT/SIT instruments. We acknowledge the use of data from the STEREO and ACE missions. The STEREO/SIT data were obtained from http://www.srl.caltech.edu/STEREO/ Level1/SIT_public.html, while the STEREO/PLASTIC and STEREO/MAG data were obtained from http://aten. igpp.ucla.edu/forms/stereo/level2_ plasma_and_magnetic_field.html. The ACE/ULEIS, ACE/SWEPAM, and ACE/MAG data were obtained from the ACE Science Center (http://www. srl.caltech.edu/ACE/ASC/level2/). G.L. and L.Z. were supported in part by NSF grant AGS1135432; NASA grants NNX13AE07G and NNX14AC08G at UAH; R.W.E., M.I.D., and M.A.D. at SWRI were supported in part by NASA grant NNX13AE07G, NSF grant AGS-1460118, and an internal research grant from Southwest Research Institute. The work at APL was supported by NSF grant AGS-0962653, contract SA4889-26309, and NASA grant NNX13AR20G from the University of California Berkeley. The work at ShangDong University was supported by NNSFC grants 41274175 and 41331068.

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The length of AB is given by r𝜃 , and the length of BD is given by dr∕ cos 𝛽 (in the right triangle BCD), where dr = ADsin𝛼 (in the right triangle ACD). We, therefore, get AD =

r𝜃 cos 𝛼 (1 + tan 𝛼 tan𝜓). 1 − (sin 𝛼∕ cos 𝜓)2

(B2)

In the triangle of ACE and ACD, the relation between tan𝛼 and tan𝜓 can be expressed as V dt CD = sh , AC AC Vsw dt CE = , cot𝜓 = AC AC

tan𝛼 =

(B3)

where Vsh and Vsw is the shock speed and fast solar wind speed and dt is time for the Sun to rotate by 𝜃 degree. Equation (B3) can be simplified into tan𝛼 tan𝜓 = Vsh ∕Vsw .

References Acuña, M. H., D. Curtis, J. L. Scheifele, C. T. Russell, P. Schroeder, A. Szabo, and J. G. Luhmann (2008), The STEREO/IMPACT magnetic field experiment, Space Sci. Rev., 136, 203–226. Agueda, N., D. Lario, E. C. Roelof, and B. Sanahuja (2005), Modeling the effects of pitch-angle scattering processes on the transport of solar energetic particles along the interplanetary magnetic field, Adv. Space Res., 35, 579–585, doi:10.1016/j.asr.2005.04.069. Bruno, R., and V. Carbone (2013), The solar wind as a turbulence laboratory, Living Rev. Sol. Phys., 10(2013), 2–208. Buˇcík, R., U. Mall, A. Korth, and G. M. Mason (2009), On acceleration of